In a head dip bungee jump from abridge over a river, the bungee cord is fastened to the jumpers ankles. The jumper then steps off and falls towards the river until the cord becomes taut. At that point, the cord begins to slow the jumper's descent, until his head just touches the water. The bridge is 22.0m above the river. The un-stretched length of the cord id 12.2m. The jumper is 1.80m tall and has a mass of 60kg. Determine the:
a) The required value of the spring constant for the jump to be successful.
b) The acceleration of the jumper at the bottom of the descent.
The gravitational potential energy of the jumper is mgh, where m is 60kg and h is 22m - 1.8m or 20.2m.
The PE of the bungee at full length is 1/2kx^2, where x is the difference between stretched and unstretched length, i.e. 20.2m - 12.2m or 8m.
Setting these to equal: 1/2kx^2=mgh then k = 2mgh/x^2
k= 2*60*9.8*20.2/8^2= 371N/m
To determine the required value of the spring constant for the jump to be successful, we need to consider the forces acting on the jumper during the descent.
a) The first step is to find the total length of the stretched bungee cord when the jumper's head just touches the water. To do this, we subtract the un-stretched length of the cord from the distance between the bridge and the water's surface:
Total length of stretched bungee cord = Bridge height + underwater distance
Total length of stretched bungee cord = 22.0m + 1.80m
Total length of stretched bungee cord = 23.80m
Next, we need to calculate the extension of the bungee cord. The extension of the cord is the difference between the total length of the stretched cord and the un-stretched length of the cord:
Extension of bungee cord = Total length of stretched cord - Un-stretched length of cord
Extension of bungee cord = 23.80m - 12.2m
Extension of bungee cord = 11.60m
The next step is to calculate the force exerted by the bungee cord at maximum extension. We can use Hooke's Law, which states that the force exerted by an ideal spring is directly proportional to its extension:
Force = Spring constant * Extension
Force = k * x
Where:
Force is the force exerted by the bungee cord
Spring constant is the value we want to find
Extension is the extension of the bungee cord
Since the bungee cord is attached to the jumper's ankles, the force exerted by the cord will counteract the gravitational force pulling the jumper downwards.
Force exerted by bungee cord = Jumper's mass * gravitational acceleration
Force exerted by bungee cord = 60kg * 9.8m/s^2
Force exerted by bungee cord = 588N
Setting the force exerted by the bungee cord equal to the force calculated using Hooke's Law:
588N = k * 11.60m
Solving for k:
k = 588N / 11.60m
k ≈ 50.69 N/m
Therefore, the required value of the spring constant for the jump to be successful is approximately 50.69 N/m.
b) To find the acceleration of the jumper at the bottom of the descent, we can use Newton's second law of motion:
Force = mass * acceleration
Force exerted by bungee cord - gravitational force = mass * acceleration
588N - (60kg * 9.8m/s^2) = 60kg * acceleration
588N - 588N = 60kg * acceleration
0N = 60kg * acceleration
Since the net force is zero at the bottom of the descent, the acceleration of the jumper is zero.
Therefore, the acceleration of the jumper at the bottom of the descent is 0m/s^2.
To solve this problem, we'll need to consider the forces acting on the jumper at different stages of the jump.
a) The required value of the spring constant for the jump to be successful:
1. First, let's calculate the distance the jumper will fall until the cord becomes taut. The un-stretched length of the cord is 12.2m, and the bridge is 22.0m high. Therefore, the distance the cord stretches during this fall is:
Total distance fallen = 22.0m - 12.2m = 9.8m
2. The cord acts like a spring, exerting a force on the jumper as it stretches. According to Hooke's law, the force exerted by a spring is proportional to the extension (Δx) of the spring: F = kΔx, where k is the spring constant.
3. To determine the required value of the spring constant, we need to calculate the extension of the cord. Since the cord stretches until it becomes taut, the extension is equal to the initial length of the cord (12.2m).
4. Using the equation F = kΔx, we can rearrange it to solve for the spring constant:
k = F / Δx,
where F is the force acting on the jumper and Δx is the extension of the spring.
5. The force acting on the jumper is the weight of the jumper, which can be calculated using the formula: F = mg, where m is the mass of the jumper (60kg) and g is the acceleration due to gravity (approximately 9.8m/s²).
6. Now we can substitute the values into the equation to find the spring constant:
k = mg / Δx = (60kg)(9.8m/s²) / 12.2m
b) The acceleration of the jumper at the bottom of the descent:
1. The acceleration of the jumper can be calculated using Newton's second law of motion: F = ma, where F is the net force acting on the jumper and a is the acceleration.
2. At the bottom of the descent, the only force acting on the jumper is the tension in the cord, which is pulling the jumper upward.
3. The magnitude of the tension force can be found using Hooke's law: F = kΔx, where F is the force, k is the spring constant, and Δx is the extension of the cord.
4. At the bottom of the descent, the extension of the cord is equal to the height of the jumper (1.80m).
5. Substituting the values into the equation, we have:
F = kΔx = k(1.80m)
6. Since the tension force is acting upward, it is equal in magnitude but opposite in direction to the force of gravity acting downward.
7. Therefore, the net force acting on the jumper is the difference between the tension force and the force of gravity:
Net force = Tension - Weight = Tension - mg
8. Equating the net force to the product of the mass and acceleration, we get:
ma = Tension - mg
9. Rearranging the equation to solve for the acceleration, we have:
a = (Tension - mg) / m
10. Now we can substitute the values (spring constant and mass) into the equation to calculate the acceleration of the jumper at the bottom of the descent.
By following these steps, you should be able to determine the required value of the spring constant (a) and the acceleration of the jumper at the bottom of the descent (b) in the given bungee jump scenario.